Dimension Reduction for Model-based Clustering via Mixtures of ...

Chapter 2BackgroundFinite mixtures of distributions provide a mathematical approach for fitting statisticalmodels to a wide variety of random phenomena. McLachlan and Basford (1988) andMcLachlan and Peel (2000) give an extensive description of finite mixture models whichhave become increasingly popular due to their flexibility. The most popular applicationof these models occurs for scenarios where data exhibit group structure or where thedata can be investigated for such structure.A p-dimensional random vector X is said to arise from a parametric finite mixturedistribution if ∀x ⊂ X, one can writep(x|ϑ) =G∑π g p g (x|θ g ) ,g=1where G is the number of components, π g are mixing proportions such thatG∑π g = 1 and π g > 0 ,g=1and ϑ = (π 1 , . . . , π G , θ 1 , . . . , θ G ) is the parameter vector. The p g (x|θ g ) are called componentdensities.3